Section 5.1 The Kinematic Screw
Recall from
Section 3.6 that the velocity field
\(P\in\cB \mapsto \vel_{P\in\cB /\cA}\) of a rigid body
\(\cB\) in motion relative to rigid body (or referential)
\(\cA\) satisfies the property
\(\vel_{Q \in\cB / \cA}= \vel_{P\in\cB / \cA} + \bom_{\cB /\cA} \times \br_{PQ}\text{.}\)
Definition 5.1.1. Kinematic screw.
The velocity field \(P\in\cB \mapsto \vel_{P\in\cB /\cA}\) defines a screw, called kinematic screw of \(\cB\) relative to \(\cA\text{.}\) It is denoted as
\begin{equation}
\{ \cV _{\cB / \cA} \} = \left\{
\begin{array}{c}
\bom_{\cB/\cA} \\
\vel_{P \in\cB /\cA}
\end{array}
\right\}\tag{5.1.1}
\end{equation}
The “resultant” of screw \(\{ \cV _{\cB / \cA} \}\) is angular velocity \(\bom_{\cB/\cA}\text{.}\) Its ``moment’’ about point \(P\) is velocity \(\vel_{P\in\cB / \cA}\text{.}\) Recall that the velocity of a point \(P\) is denoted \(\vel_{P\in\cB / \cA}\) to ensure that the velocity \(P\) is viewed as that of a point of \(\cB\text{.}\)
The properties stated below follow immediately from the general properties of screws.
Corollary 5.1.2.
1. Equiprojectivity: The velocity field \(P\in\cB \mapsto \vel_{P\in\cB /\cA}\) is equiprojective, that is, \(\vel_{P\in\cB /\cA} \cdot \br_{PQ} = \vel_{Q \in\cB / \cA} \cdot \br_{PQ}\) for any two points attached to \(\cB\text{.}\)
This implies that the projections of the velocities of \(P\) and \(Q\) onto the line joining \(P\) and \(Q\) are identical.
2. Scalar invariant: The scalar quantity \(\bom_{\cB/\cA} \cdot \vel_{P\in\cB /\cA}\) is an invariant, that is, it is independent of the chosen point \(P\) of \(\cB\text{.}\)
3. Instantaneous screw axis: At any instant in time, whenever \(\bom_{\cB/\cA}\neq \bze\text{,}\) the kinematic screw \(\{\cV _{\cB / \cA} \}\) is characterized by an axis \(\Delta_{\cB/\cA}\) defined as the set of points whose velocity is collinear to the angular velocity \(\bom_{\cB/\cA}\text{.}\) The velocity field takes the same values along \(\Delta_{\cB/\cA}\text{,}\) that is, \(\vel_{Q\in\Delta /\cA} = p \,\bom_{\cB/\cA}\text{,}\) where the pitch \(p\) is the (invariant) parameter given by
\begin{equation}
p = { \bom_{\cB/\cA} \cdot \vel_{P \in\cB /\cA} \over \bom_{\cB / \cA}^2 } \tag{5.1.2}
\end{equation}
Axis \(\Delta_{\cB/\cA}\) (or simply \(\Delta\)) is referred to as the instantaneous screw axis of \(\cB\) (relative to \(\cA\)).
